Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm^3 What's the volume of a sphere if its radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter?
A.) 54 cm^3
В.) 12 сm^3
С.) 18 сm^3
D.) 24 cm^3
First, we need to find the radius and height of the cylinder. Since the volume of the cylinder is 36 cm^3 and the formula for the volume of a cylinder is V = πr^2h, we can set up the following equation:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, which is twice the radius of the sphere, we can write h = 2r. Substituting this into the equation above, we get:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = ∛(18/π)
r ≈ 1.64 cm
Now that we have found the radius of the cylinder, we can use it to find the volume of the sphere. The formula for the volume of a sphere is V = (4/3)πr^3. Substituting the radius we found earlier, we get:
V = (4/3)π(1.64)^3
V ≈ 18 cm^3
Therefore, the closest answer choice is C.) 18 cm^3.